On the variety associated to the ring of theta constants in genus 3

Due to fundamental results of Igusa and Mumford the N = 2(g-1 )(2(g) + 1) theta constants of first kindSigma(n integral) exp pi i(Z[n + a/2] + 2b' (n + a/2)), a,b integral.define for each genus g an injective holomorphic map of the Satake compactification X-g(4, 8) = (H-g/Gamma(g)[4,8]) over bar into the projective space PN-1. Moreover, this map is biholomorphic onto the image outside the Satake boundary. It is not biholomorphic on the whole in the cases g >= 6. Igusa also proved that in the cases g <= 2 this map is biholomorphic onto the image. In this paper we extend this result to the case g = 3. So we show that the theta mapX-3(4,8) -> P-35is biholomorphic onto the image. This is equivalent to the statement that the image is a normal subvariety of P-35.